Linear Fractional Composition Operators Research Articles

Frequent hypercyclicity of the tensor products of composition operators on weighted Dirichlet spaces

In this paper, we mainly study the frequent hypercyclicity of a scalar multiple of the tensor product of two linear fractional composition operators on , where is the weighted Dirichlet space. We will restrict the discussion to the situations that both φ and ψ are hyperbolic non automorphisms, hyperbolic automorphisms and parabolic automorphisms, respectively. Meanwhile, we give the equivalent conditions of to be frequently hypercyclic, hypercyclic, mixing and chaotic.

Components of linear fractional composition operators on spaces of holomorphic functions

In this paper we consider the topological structure problem for the space C(X) of all composition operators on a space X of holomorphic functions over the unit disc, where X is one of the following function spaces: the Hardy space H2, Bergman spaces Aαp, the space H∞, weighted Banach spaces Hv with sup-norm, the classical Bloch space B, and weighted Bloch spaces Bv. In particular, we give a necessary and sufficient condition for two linear fractional composition operators to be in the same component of C(X). A characterization of isolated such composition operators in C(X) is also established.

Recurrence of multiples of composition operators on weighted Dirichlet spaces

A bounded linear operator T acting on a Hilbert space $$\mathcal {H}$$ is said to be recurrent if for every non-empty open subset $$U\subset \mathcal {H}$$ there is an integer n such that $$T^n (U)\cap U\ne \emptyset$$ . In this paper, we completely characterize the recurrence of scalar multiples of composition operators, induced by linear fractional self maps of the unit disk, acting on weighted Dirichlet spaces $$\mathcal {S}_\nu$$ ; in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete previous work of Costakis, Manoussos, and Parissis on the recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples $$(\lambda ,\nu ,\phi )\in {\mathbb {C}}\times \mathbb {R}\times \mathrm{LFM}(\mathbb {D})$$ for which the scalar multiple of composition operator $$\lambda C_\phi$$ acting on $$\mathcal {S}_\nu$$ fails to be recurrent.

Universal composition operators

A Hilbert space operator U is called \textit{universal} (in the sense of Rota) if every Hilbert space operator is similar to a multiple of U restricted to one of its invariant subspaces. It follows that the \textit{invariant subspace problem} for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for U are one dimensional. In this article we characterize all linear fractional composition operators Cϕf=f∘ϕ that have universal translates on both the classical Hardy spaces H2(C+) and H2(D) of the half-plane and the unit disk, respectively. The new example here is the composition operator on H2(D) with affine symbol ϕa(z)=az+(1−a) for $0

Adjoints of Composition Operators with Linear Fractional Symbols on the Dirichlet Space of the Disk or the Ball

In this paper, we discuss the relationship between the adjoints of linear fractional composition operators with respect to two equivalent norms on the Dirichlet space of the disk or the ball. We give a simple and insightful connection between these two norms using the backward shift and the Volterra integration operator. More explicit formulas for the adjoints of linear factional composition operators on the Dirichlet space of the disk or the ball are obtained.

SPECTRA OF LINEAR FRACTIONAL COMPOSITION OPERATORS ON THE GROWTH SPACE AND BLOCH SPACE OF THE UPPER HALF-PLANE

In this article, we provide a complete description of the spectra of linear fractional composition operators acting on the growth space and Bloch space over the upper half-plane. In addition, we also prove that the norm, essential norm, spectral radius and essential spectral radius of a composition operator acting on the growth space are all equal.

Linear Fractional Composition Operators Over the Half-Plane

In the setting of the Hardy spaces or the standard weighted Bergman spaces over the unit ball in $${\mathbf {C}}^n$$ , linear fractional composition operators are known to behave quite rigidly in the sense that they cannot form any nontrivial compact differences or, more generally, linear combinations. In this paper, in the setting of the standard weighted Bergman spaces over the half-plane, we completely characterize bounded/compact differences of linear fractional composition operators. Our characterization reveals that a linear fractional composition operator can possibly form a compact difference, which is a new half-plane phenomenon due to the half-plane not being bounded. Also, we obtain necessary conditions and sufficient conditions for a linear combination to be bounded/compact. As a consequence, when the weights and exponents of the weighted Bergman spaces are restricted to a certain range, we obtain a characterization for a linear combination to be bounded/compact. Applying our results, we provide an example showing a double difference cancellation phenomenon for linear combinations of three linear fractional composition operators, which is yet another half-plane phenomenon.

Adjoints of linear fractional composition operators on weighted Hardy spaces

It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted Bergman space $A^2_{\alpha}(\mathbb{D})$ over the unit disk, the adjoint of a linear fractional composition operator equals the product of a composition operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of analytic functions on the disk whose first derivatives belong to $H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal of compact operators. In this paper we investigate what the situation is like on other weighted Hardy spaces.

Spectra of linear fractional composition operators on the Hardy and weighted Bergman spaces of the half-plane

We compute the spectra and the essential spectra of bounded linear fractional composition operators acting on the Hardy and weighted Bergman spaces of the upper half-plane. We are also able to extend the results to weighted Dirichlet spaces of the upper half-plane.

The spectra of linear fractional composition operators on weighted Dirichlet spaces

We completely determine the spectra of composition operators induced by linear fractional self-maps of the unit disc acting on weighted Dirichlet spaces; extending earlier results by Higdon [8] and answering the open questions in this context.

Dynamics of composition operators on weighted Bergman spaces

We discuss the frequent hypercyclicity and hypercyclicity of multiples of the linear fractional composition operators on Aαp and give equivalent conditions of the frequent hypercyclicity and hypercyclicity. We also characterize disjoint hypercyclicity and disjoint mixing behavior of finitely many linear fractional composition operators on the weighted Bergman spaces Aαp.

Disjoint mixing linear fractional composition operators in the unit ball

In the present paper, we investigate the disjoint mixing property of finitely many linear fractional composition operators acting on the space of holomorphic functions on the unit ball in CN, and generalize parts of the results obtained by Bès, Martin and Peris in 2011.

Spectra of linear fractional composition operators on H2 (BN)

We characterize the spectra of composition operators on the Hardy space , when their symbols are elliptic or hyperbolic linear fractional self-maps of the unit ball . Therefore, combining with the result obtained by Bayart, the spectra of all linear fractional composition operators on are completely determined.

Essential normality for certain finite linear combinations of linear-fractional composition operators on the Hardy space H 2

In 1999 Nina Zorboska and in 2003 P. S.Bourdon, D. Levi, S.K.Narayan and J.H. Shapiro investigated the essentially normal composition operator $${C_\varphi }$$ , when φ is a linear-fractional self-map of D. In this paper first, we investigate the essential normality problem for the operator T w $${C_\varphi }$$ on the Hardy space H 2, where w is a bounded measurable function on ∂D which is continuous at each point of F(φ), φ ∈ S(2), and T w is the Toeplitz operator with symbol w. Then we use these results and characterize the essentially normal finite linear combinations of certain linear-fractional composition operators on H 2.

Adjoints of linear fractional composition operators on [formula omitted

Given φ:D→D, an analytic self-map of the unit disc in C, the composition operator Cφ is defined by Cφ(f)=f∘φ for f belonging to some Hilbert space of analytic functions on D. We present a formula for the adjoint of linear-fractionally induced composition operators on S2(D), a weighted Hardy space, which consists of analytic functions on the disc whose first derivative is in H2(D).

Compact differences of composition operators on Holomorphic Function Spaces in the unit ball

We find a lower bound for the essential norm of the difference of two composition operators acting on H 2(B N) or A 2 s (B N) ( s > −1). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.

Disjoint hypercyclic linear fractional composition operators

We characterize disjoint hypercyclicity and disjoint supercyclicity of finitely many linear fractional composition operators acting on spaces of holomorphic functions on the unit disc, answering a question of Bernal-González. We also study mixing and disjoint mixing behavior of projective limits of endomorphisms of a projective spectrum. In particular, we show that a linear fractional composition operator is mixing on the projective limit of the S v spaces strictly containing the Dirichlet space if and only if the operator is mixing on the Hardy space.

Essential normality of linear fractional composition operators in the unit ball of ℂ N

Let φ be a linear fractional self-map of the ball BN with a boundary fixed point e1, we show that $$ 1 - \operatorname{Re} \phi _1 \left( z \right) \sim \operatorname{Re} \left( {1 - z_1 } \right) $$ holds in a neighborhood of e1 on \( {B_N } \). Applying this result we give a positive answer for a conjecture by MacCluer and Weir, and improve their results relating to the essential normality of composition operators on H2(BN) and Aγ2 (BN) (γ > −1). Combining this with other related results in MacCluer & Weir, Integral Equations Operator Theory, 2005, we characterize the essential normality of composition operators induced by parabolic or hyperbolic linear fractional self-maps of B2. Some of them indicate a difference between one variable and several variables.

Linear fractional composition operators on the Dirichlet space in the unit ball

We investigate the adjoints of linear fractional composition operators Cϕ acting on classical Dirichlet space D(BN) in the unit ball BN of ℂN, and characterize the normality and essential normality of Cϕ on D(BN) and the Dirichlet space modulo constant function D0(BN), where ϕ is a linear fractional map ϕ of BN. In addition, we also show that for any non-elliptic linear fractional map ϕ of BN, the composition maps σ o ϕ and ϕ o σ are elliptic or parabolic linear fractional maps of BN.

Norms and spectral radii of linear fractional composition operators on the ball

We give a new proof that every linear fractional map of the unit ball induces a bounded composition operator on the standard scale of Hilbert function spaces on the ball, and obtain new norm bounds analogous to the standard one-variable estimates. We also show that Cowen's one-variable spectral radius formula extends to these operators. The key observation underlying these results is that every linear fractional map of the ball belongs to the Schur–Agler class.